2 Answers
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There are only two situations where that phrase should apply: If you are a baby, or engaged in sexual relations. It seems likely that this means: Do not rely upon your sexual partners to be entirely honest with you. Given the amount of treachery (and sex) in Game of Thrones, this may be good advice.

Will had known they would drag him into the quarrel sooner or later. He wished it had been later rather than sooner. "My mother told me that dead men sing no songs," he put in.

"My wet nurse said the same thing, Will," Royce replied. "Never believe anything you hear at a woman's tit. There are things to be learned even from the dead." His voice echoed, too loud in the twilit forest.

"We have a long ride before us," Gared pointed out. "Eight days, maybe nine. And night is falling."

I think it's sort of a double entendre. Literally, it means you shouldn't believe what you hear from the women who raised you. (They might be repeating old wives' tales.) But the sentence could also refer to sexual partners.

Ser Waymer Royce is young and arrogant. Like many of the characters in A Game of Thrones, he probably doesn't have much respect for women in general. His words are more dismissive than anything else. He wants to portray himself as courageous and manly, not like a (weak, scared) woman. As you will shortly find out, this does not end well for him.